If the codomain of the functions under consideration have a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain.[2]

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Let X and Y be two topological spaces, and let C(X, Y) denote the set of all continuous maps between X and Y. Given a compact subsetK of X and an open subsetU of Y, let V(K, U) denote the set of all functions f ∈ C(X, Y) such that f (K) ⊂ U. Then the collection of all such V(K, U) is a subbase for the compact-open topology on C(X, Y). (This collection does not always form a base for a topology on C(X, Y).)

When working in the category of compactly generated spaces, it is common to modify this definition by restricting to the subbase formed from those K which are the image of a compactHausdorff space. Of course, if X is compactly generated and Hausdorff, this definition coincides with the previous one. However, the modified definition is crucial if one wants the convenient category of compactly generated weak Hausdorff spaces to be Cartesian closed, among other useful properties.[3][4][5] The confusion between this definition and the one above is caused by differing usage of the word compact.

If * is a one-point space then one can identify C(*, Y) with Y, and under this identification the compact-open topology agrees with the topology on Y. More generally, if X is a discrete space, then C(X, Y) can be identified with the cartesian product of |X| copies of Y and the compact-open topology agrees with the product topology.

If X is Hausdorff and S is a subbase for Y, then the collection {V(K,U) : U ∈ S} is a subbase for the compact-open topology on C(X, Y).[6]

If Y is a metric space (or more generally, a uniform space), then the compact-open topology is equal to the topology of compact convergence. In other words, if Y is a metric space, then a sequence{ fn } converges to f in the compact-open topology if and only if for every compact subset K of X, { fn } converges uniformly to f on K. In particular, if X is compact and Y is a uniform space, then the compact-open topology is equal to the topology of uniform convergence.

If Y is a locally compact Hausdorff (or preregular) space, then the evaluation map e : C(Y, Z) × Y → Z, defined by e( f , x) = f (x), is continuous. This can be seen as a special case of the above where X is a one-point space.

If X is compact, and Y is a metric space with metric d, then the compact-open topology on C(X, Y) is metrisable, and a metric for it is given by e( f , g) = sup{d( f (x), g(x)) : x in X}, for f , g ∈ C(X, Y).

Let X and Y be two Banach spaces defined over the same field, and let C m(U, Y) denote the set of all m-continuously Fréchet-differentiable functions from the open subset U ⊆ X to Y. The compact-open topology is the initial topology induced by the seminorms